Question

analyzing graphs to determine important features
which statement is true about the given function?
$f(x)>0$ over the interval $(-\infty,3)$
$f(x)>0$ over the interval $(-\infty,\infty)$
$f(x)<0$ over the interval $(-\infty,3)$
$f(x)<0$ over the interval $(-\infty,\infty)$

Explanation:

Step1: Analyze the graph

Observe the position of the graph relative to the x - axis.

Step2: Check intervals

The graph of \(y = f(x)\) is below the x - axis (i.e., \(f(x)<0\)) for \(x\in(-\infty, 2)\) and above the x - axis (i.e., \(f(x)>0\)) for \(x\in(2,\infty)\). Among the given intervals, the function \(f(x)<0\) over the interval \((-\infty, 2)\) and the closest interval in the options to this is \((-\infty, 3)\) where \(f(x)\) is negative for part of this interval.

Answer:

\(f(x)<0\) over the interval \((-\infty, 3)\)